What is the difference between a theorem, a lemma, and a corollary?

I prepared the following handout for my Discrete Mathematics class (here’s a pdf version).

Definition — a precise and unambiguous description of the meaning of a mathematical term. It characterizes the meaning of a word by giving all the properties and only those properties that must be true.

Theorem — a mathematical statement that is proved using rigorous mathematical reasoning. In a mathematical paper, the term theorem is often reserved for the most important results.

Paradox — a statement that can be shown, using a given set of axioms and definitions, to be both true and false. Paradoxes are often used to show the inconsistencies in a flawed theory (Russell’s paradox). The term paradox is often used informally to describe a surprising or counterintuitive result that follows from a given set of rules (Banach-Tarski paradox, Alabama paradox, Gabriel’s horn).

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When I teach our Intro to Proof course, one assignment that I give is to have students write a story that personifies each of these terms (except Identity and Paradox, but maybe I’ll add them). For example, maybe Theorem is a General and Corollary is a Lieutenant. The idea is to use creative writing to explore the nuanced distinctions between similar terms: When is something a Theorem versus a Proposition?

The stories are really fun to read, and sometimes lead to other points. For example, when people give genders to the terms, Theorem is overwhelmingly male and Corollary is almost always female. I don’t mention this ahead of time, but if I notice it when I read the papers I point it out when I hand them back. [No commentary — just the observation. Most of our math majors are women, and this observation seems to be rather powerful.]

To begin with, a conjecture is not necessarily a truism. Fundamentally it is a phrase expressing an accepted or an accommodated belief. Does this matter? Greatly. For what does it mean something to be true?. It means that it can be proven through a logical process of deduction starting with a set of (atomic) axioms. However, there are mathematical constructs that “live” outside this process of deduction, e.g. the Axiom of Choice (A of C) which is presumed (not assumed!) a priori, and even more broadly, those constructs that embody Godel’s 1st Incompleteness Theorem (G1IT), which arise (arguably) a posteriori. Putting aside the issues with A of C for a moment, G1IT shows there are constructs that can not be proven one way or the other and yet still be consistent and factual. That there is no a posteriori way to determine these constructs, but yet we can say they do exist. Note this is not the same thing as the A of C which is considered to be true a priori, but also not provable. In any case, truth in G1IT is not the same thing as we use it in the arena of axiomatic deduction. Futhermore and back to A of C for a moment, does this mean that the Axiom of Choice, is not an axiom but a postulate, or [gulp] a conjecture? This segues to my next point.

In generally I have taught in the past, definitions are tied together by propositions to enable the creation of axioms. Axioms are the building blocks to construct conjectures and claims, which feed our need to discover, explore and investigate. If these conjectures and claims are to be accepted a priori they become postulates. However if a posteriori process of deduction is used (this means also mathematically) then we have the lemma->theorem->corollary chain.
And here belies the another issue. Depending on your deductive approach, what one mathematician may call a theorem (even if it has been generally accepted for hundreds of years by most of the mathematicians), to another it maybe a lemma, or even a corollary. In essence, the well known Theorems today, and this goes for Lemmas and Corollaries also, have this tag placed on them for historical reasons, and nothing more. Personally I do consider this to be a dangerous course of endeavor for it forces students to think a certain linear way. The connotations of brainwashing are enliven here. But I digress, and leave this sensitive topic for another time and place.

Personally, and this is only a suggestion, for of course, there is much contention on many of the above points I have raised, I believe you should add into your list, the concepts of a priori and a posteriori. As I have outlined, there are “things” in this Universe, as I would like to refer to it as Galileo’s Universe, that we can tangibly prove and or disprove, these truisms are all deduced a posteriori. However, we do have constructs that have arguably no true or false understanding as we can express in Galileo’s Universe, but live in a larger Platonean Universe. Thus how can you call such things a conjecture, postulate or whatever?

I can’t help thinking that 100 years ago Hilbert and his contemporaries were having much the same argument. Here we are 100 years later, and still people are trying to use words in the same way that Hilbert was trying to build his program for a complete, consistent, sound mathematics. Thanks to Godel, we now know that a Hilbert Mathematics is impossible, so why are we 100 years later pushing a Hilbert Mathematical English?

I note lastly that I know that you are trying to give a simple definition to these terms, but as I outlined, there belies the danger. I truly believe if your going to give definitions, you better also show how that definitions or approaches are not written in stone, and that there are many perspectives where the definitions you give do not fit in with what is really evident, even though 90% of the school of thought takes the same easy way out. It just perpetuates the bad mathematical understanding. And mathematics really requires a flexible mind, and not a rigged linear thought process (which is how it is, sadly, mostly taught today)… Godel should be proof of that!

Thank you for your thoughtful response to my post. I completely agree that things are MUCH more subtle than what I wrote. However, I had to keep my audience in mind—this is a group of first semester freshman or first semester sophomores taking their first proof-writing course. Some of these words were brand new to them. It would have been completely inappropriate for me to go into the foundations of mathematics with them. Many of these students will continue on to major in mathematics and we’ll have other opportunities to discuss the finer points.

Freshman or first years, it matters not. You start teaching false understandings at the beginning of a mathematical education, you end up with just confused and poor mathematicians. Worse yet, you end up with poor engineers and ridiculous physics theories, which is what we have today. Disciplines like Physics are in trouble, this has been so now for the better part of 30 years. There are subtleties that need to be understood, especially now in physics, and these subtleties are being ignored. There is a fundamental problem how we do theoretical physics and the way we interpret mathematics in physics, that is, how we do mathematical physics. I am of a school where we should be doing physical mathematics – that is, learn from observation and let reality guide your mathematics – this is contrary to mathematical physics, where the math is pushing how we want reality. That is why I gave the argument above on the interpretation of what is a theory, lemma or corollary. Perspective is illusory, not mathematical fact.

As a small but very profound illustration, consider how the Hamiltonian is calculated for a particle in a box, or infinite well (it matters not the case). After some effort, the wave functions are determined for the possible states (energy, position, momentum, whatever) that the particle is allowed. But is this really true! Consider what the mathematics is really telling us. We have calculated these possible quantized values for the entire system, in situ and a priori (refer back to my initial commentary for why this is important). So the question begs, how does a particle get to know the entire system, and know how to go into these appropriate quantum states IMMEDIATELY. The mathematics, is insitu! Reality is not. You can bring Relativity into this, it doesn’t change anything, the problem still remains, its just one level higher. This problem becomes self evident once you get into entanglement, but even there, the mathematics we use breaks down.

I’ll end by giving you a first year dilemma, since you are so focused on freshmans and their capacity to understand basic mathematics. Let us see how you can explain this to them. Riddle me this, the inner dot product of two vectors is considered to be a scalar, whereas the cross product is vector. Furthermore, the inner dot product is interpreted by every mathematician and physicist in every university in the world today, as a projection of one vector upon the other, and the cross product illustrates the area (if we take units of distance) of two vectors. But is this true? Why is it that inner dot product has units of area, like the cross product. And if the inner dot product truly does represent an area, what area is it? Furthermore, reinterpreting the dot product to this “forgotten area”, what does this do to our interpretation of, say, Maxwell’s equations? As you can see, if you start off with a poor or misguided understanding in mathematics, it just may take you down a path to paradoxical theories, like Quantum Theory and Relativity. Paradoxes are a sign that it is not your theory that is flawed, but the axioms that the theory is supported upon is in error. Just imagine, two hundred years of the greatest mathematical minds and greatest physicists in history all missed one important and minor thing, like understanding what the dot product really is. Could this really be true? Its not like its ever happened before is it? Hmmm, two and half thousand years of believing that the Earth was the center of the known universe, does come to mind.

Just because all the sheep follow the shepherd, it doesn’t mean that the shepherd knows where he is going.

I gave you an argument earlier that linear thinking is dangerous. For it brainwashes people into thinking one way. I can tell you now, there are very deep problems and mistakes we have made in physics and mathematics, all because, we do mathematical physics and not physical mathematics. There are other ways to look at reality, and if you don’t teach them early in the educational process, you only perpetuate the lies of the past. How many centuries must past before humanity learns these lessons?

Thanks for this nice post and the follow-up discussions, I was looking for quite sometime. Why not include the dictionary meanings (giving references) of these terms to give more authenticity to the definitions!

This site is really helpful to all the students of Mathematics and the definitions you provided are somehow true but not absolutely true or true all the time. I know you considered your audience but I believe we have to consider the truth first before anything else. You may go with the definitions that you have but you must not forget to remind your students that the world of Mathematics is not linear and can be anything they think as long as they can prove it just like what other Mathematics lovers are saying, relax…prove a theorem. ;)

I’m a K-12 educator – not a math teacher but mathematics was a favorite subject of mine as a student – and I love the idea of creative writing to explore ideas in math and other subjects. I’m going to share this idea with math teachers I know.

Other potentially interesting ways to explore the relationships between these statements would be [1] Venn diagrams and [2] concept maps. I’m surely not the first person to think of this so I’m going to look and see if anyone has published something like this on the intertubes.

(Semi-related: an English teacher used a spreadsheet to graph the relative importance of main characters in Romeo and Juliet – I think every kid in that class will have unique memories and understandings of R&J as a result. The point is that using non-traditional means to explore learning can be very cool and illuminating).

I’m not a math teacher (currently a student), but I am a tutor for many of my friends and peers in the more abstract or proof-based mathematics. I’d like to know if my explanation of a Lemma vs a Theorem would be correct.

Usually, I start by saying that a Theorem is something that may or may not seem likely, but can be proven by logical building blocks, postulates and axioms, as well as other theorems. When working on a theorem, there are other things that need to be proven inside of it to continue on, and I label those as “Lemmas”. I like to say that I only call something a Lemma when I’m only using it within a theorem, and, if I were to simply present it without the Theorem we’re trying to prove, I would call them theorems because Lemmas are, themselves, statements that have been proven true.

Sometimes I use the analogy that we use tools (postulates/axioms) to build stuff, like a a processor, motherboard, and other components for a computer (I call them Lemmas), and we put them together to make a Computer (our theorem). However, our theorem, the computer, may just be another lemma if we look at a greater scale, like a computer network, that uses multiple computers and even other, new components (ethernet cables, routers, etc) all put together. Would this analogy be correct to distinguish between a theorem and a lemma (as a matter of scope depending on our end result)?
Most of the time, this discussion comes from proving the relationship between the dot product of two vectors with the cosine of the angle between them, and I prove that v dot v = ||v||^2. In this context, I call it a lemma but I also tell them that, if we weren’t looking to prove that relationship between angles and dot products, I would call it an interesting property that we’d need to prove (and, thus, a theorem).

I would like to be teacher in higher mathematics, so any advice and/or criticism is welcome.